Consecutive Numbers in Equilateral Triangle
Source: India IMO Training Camp 2016, Practice 2, Problem 3
July 22, 2016
combinatorics
Problem Statement
An equilateral triangle with side length is divided into congruent triangular cells as shown in the figure below. Initially all the cells contain . A move consists of selecting two adjacent cells (i.e., cells sharing a common boundary) and either increasing or decreasing the numbers in both the cells by simultaneously. Determine all positive integers such that after performing several such moves one can obtain consecutive numbers in some order.
[asy] size(3cm);
pair A=(0,0),D=(1,0),B,C,E,F,G,H,I;
G=rotate(60,A)*D;
B=(1/3)*D; C=2*B;I=(1/3)*G;H=2*I;E=C+I-A;F=H+B-A;
draw(A--D--G--A^^B--F--H--C--E--I--B,black);[/asy]